Answer
$(x+\frac{1}{2})^2+\frac{3}{4}$
See graph.
Work Step by Step
Step 1. $f(x)=x^2+x+1=x^2+x+\frac{1}{4}+\frac{3}{4}=(x+\frac{1}{2})^2+\frac{3}{4}$
Step 2. To graph $f(x)$, start with $y=x^2$, shift $\frac{1}{2}$ unit to the left, then shift $\frac{3}{4}$ unit up. See graph.